Limits, Sequences, and Sums

2021 ◽  
pp. 28-50
Author(s):  
James Davidson
Keyword(s):  
Regular Variation ◽  
Real Variable ◽  
Cauchy Sequences ◽  
Order Of Magnitude ◽  
Cluster Points

This chapter first introduces the concepts of real sequences, Cauchy sequences, limits, and cluster points. The chapter then discusses functions of a real variable, continuity concepts, convergence concepts for sequences of functions, summability and order of magnitude relations, inequalities for sums, regular variation, and limit concepts for arrays.

scholarly journals On Lacunary Mean Ideal Convergence in Generalized Randomn-Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Normed Space ◽  
Complex Numbers ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Cauchy Sequences ◽  
Limit Points ◽  
Positive Integers ◽  
Cluster Points

An idealIis a hereditary and additive family of subsets of positive integersℕ. In this paper, we will introduce the concept of generalized randomn-normed space as an extension of randomn-normed space. Also, we study the concept of lacunary mean (L)-ideal convergence andL-ideal Cauchy for sequences of complex numbers in the generalized randomn-norm. We introduceIL-limit points andIL-cluster points. Furthermore, Cauchy andIL-Cauchy sequences in this construction are given. Finally, we find relations among these concepts.

On rough convergence of ρ-Cauchy sequence of triple sequences

Analysis ◽  
2019 ◽  
Vol 39 (4) ◽  
pp. 129-133
Author(s):  
Ayhan Esi ◽  
M. Aiyub ◽  
N. Subramanian ◽  
Ayten Esi
Keyword(s):  
Cauchy Sequence ◽  
Sequence Spaces ◽  
Cauchy Sequences ◽  
Limit Points ◽  
Cluster Points

Abstract In this paper we define and study rough convergence of triple sequences and the set of rough limit points of a triple sequence. We also investigate the relations between the set of cluster points and the set of rough limit points of Cauchy sequences of triple sequence spaces.

scholarly journals Generalized ideal convergence in intuitionistic fuzzy normed linear spaces

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 811-820 ◽  
Author(s):  
Bipan Hazarika ◽  
Vijay Kumar ◽  
Bernardo Lafuerza-Guilién
Keyword(s):  
Real Number ◽  
Normed Spaces ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Linear Spaces ◽  
Intuitionistic Fuzzy ◽  
Cauchy Sequences ◽  
Positive Integers ◽  
Intuitionistic Fuzzy Normed Spaces ◽  
Cluster Points

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In [19], Kostyrko et al. introduced the concept of ideal convergence as a sequence (xk) of real numbers is said to be I-convergent to a real number e, if for each ? > 0 the set {k ? N : |xk - e| ? ?} belongs to I. The aim of this paper is to introduce and study the notion of ?-ideal convergence in intuitionistic fuzzy normed spaces as a variant of the notion of ideal convergence. Also I? -limit points and I?-cluster points have been defined and the relation between them has been established. Furthermore, Cauchy and I?-Cauchy sequences are introduced and studied. .

scholarly journals Regular variation in a fixed-point problem for single- and multi-class branching processes and queues

2018 ◽  
Vol 50 (A) ◽  
pp. 47-61 ◽  
Author(s):  
Søren Asmussen ◽  
Sergey Foss
Keyword(s):  
Fixed Point ◽  
Regular Variation ◽  
Branching Processes ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Tail Asymptotics ◽  
Order Of Magnitude ◽  
Heavy Tailed

Abstract Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are derived under heavy-tailed conditions allowing both dependence between Q and N and the tails to be of the same order of magnitude. Similar results are derived for a K-class version with applications to multi-type branching processes and busy periods in multi-class queues.

A controller for safe, convenient operation of LaB6 emitters on electron-beam instruments

1983 ◽  
Vol 41 ◽  
pp. 408-409
Author(s):  
W. J. Abramson ◽  
H. W. Estry ◽  
L. F. Allard
Keyword(s):  
Electron Beam ◽  
Control System ◽  
Thermal Shock ◽  
Appropriate Care ◽  
Electron Guns ◽  
Tungsten Filaments ◽  
Order Of Magnitude ◽  
Main Components

LaB6 emitters are becoming increasingly popular as direct replacements for tungsten filaments in the electron guns of modern electron-beam instruments. These emitters offer order of magnitude increases in beam brightness, and, with appropriate care in operation, a corresponding increase in source lifetime. They are, however, an order of magnitude more expensive, and may be easily damaged (by improper vacuum conditions and thermal shock) during saturation/desaturation operations. These operations typically require several minutes of an operator's attention, which becomes tedious and subject to error, particularly since the emitter must be cooled during sample exchanges to minimize damage from random vacuum excursions. We have designed a control system for LaBg emitters which relieves the operator of the necessity for manually controlling the emitter power, minimizes the danger of accidental improper operation, and makes the use of these emitters routine on multi-user instruments.Figure 1 is a block schematic of the main components of the control system, and Figure 2 shows the control box.

Novel magneto-optical recording materials: Bi-substituted garnet films

1991 ◽  
Vol 49 ◽  
pp. 776-777
Author(s):  
Takao Suzuki ◽  
Hossein Nuri
Keyword(s):  
Grain Size ◽  
Amorphous Film ◽  
Nitrogen Atmosphere ◽  
Optical Recording ◽  
Garnet Film ◽  
Media Noise ◽  
Garnet Films ◽  
Order Of Magnitude ◽  
A New Technique ◽  
A Current

For future high density magneto-optical recording materials, a Bi-substituted garnet film ((BiDy)3(FeGa)5O12) is an attractive candidate since it has strong magneto-optic effect at short wavelengths less than 600 nm. The signal in read back performance at 500 nm using a garnet film can be an order of magnitude higher than a current rare earth-transition metal amorphous film. However, the granularity and surface roughness of such crystalline garnet films are the key to control for minimizing media noise.We have demonstrated a new technique to fabricate a garnet film which has much smaller grain size and smoother surfaces than those annealed in a conventional oven. This method employs a high ramp-up rate annealing (Γ = 50 ~ 100 C/s) in nitrogen atmosphere. Fig.1 shows a typical microstruture of a Bi-susbtituted garnet film deposited by r.f. sputtering and then subsequently crystallized by a rapid thermal annealing technique at Γ = 50 C/s at 650 °C for 2 min. The structure is a single phase of garnet, and a grain size is about 300A.

Atomic-level imaging of the surface structure of Ag2O

1984 ◽  
Vol 42 ◽  
pp. 656-657
Author(s):  
William Krakow
Keyword(s):  
High Vacuum ◽  
Atomic Level ◽  
Ultra High Vacuum ◽  
Research Groups ◽  
Resolution Electron ◽  
Surface Reconstructions ◽  
Order Of Magnitude ◽  
High Resolution Electron Microscope ◽  
Saturated Structure ◽  
Transmission And Reflection

In recent years electron microscopy has been used to image surfaces in both the transmission and reflection modes by many research groups. Some of this work has been performed under ultra high vacuum conditions (UHV) and apparent surface reconstructions observed. The level of resolution generally has been at least an order of magnitude worse than is necessary to visualize atoms directly and therefore the detailed atomic rearrangements of the surface are not known. The present author has achieved atomic level resolution under normal vacuum conditions of various Au surfaces. Unfortunately these samples were exposed to atmosphere and could not be cleaned in a standard high resolution electron microscope. The result obtained surfaces which were impurity stabilized and reveal the bulk lattice (1x1) type surface structures also encountered by other surface physics techniques under impure or overlayer contaminant conditions. It was therefore decided to study a system where exposure to air was unimportant by using a oxygen saturated structure, Ag2O, and seeking to find surface reconstructions, which will now be described.

Near-field scanning optical microscopy

1986 ◽  
Vol 44 ◽  
pp. 642-643
Author(s):  
E. Betzig ◽  
A. Harootunian ◽  
M. Isaacson ◽  
A. Lewis
Keyword(s):  
Near Field ◽  
Optical Microscope ◽  
Visible Radiation ◽  
Field Methods ◽  
Optical Elements ◽  
Optical Region ◽  
Order Of Magnitude ◽  
Nondestructive Imaging ◽  
Scanning Optical Microscopy ◽  
Near Field Scanning

In general, conventional methods of optical imaging are limited in spatial resolution by either the wavelength of the radiation used or by the aberrations of the optical elements. This is true whether one uses a scanning probe or a fixed beam method. The reason for the wavelength limit of resolution is due to the far field methods of producing or detecting the radiation. If one resorts to restricting our probes to the near field optical region, then the possibility exists of obtaining spatial resolutions more than an order of magnitude smaller than the optical wavelength of the radiation used. In this paper, we will describe the principles underlying such "near field" imaging and present some preliminary results from a near field scanning optical microscope (NS0M) that uses visible radiation and is capable of resolutions comparable to an SEM. The advantage of such a technique is the possibility of completely nondestructive imaging in air at spatial resolutions of about 50nm.

Detection, Averaging, and 3D Reconstruction of Biological Specimens on Hypercubes (Transputer-based) Computers

1990 ◽  
Vol 48 (1) ◽  
pp. 454-455
Author(s):  
Jose-Maria Carazo ◽  
I. Benavides ◽  
S. Marco ◽  
J.L. Carrascosa ◽  
E.L. Zapata
Keyword(s):  
3D Reconstruction ◽  
3D Structure ◽  
Three Dimensional ◽  
Software Tools ◽  
Mapping Method ◽  
Computer Architectures ◽  
Parallel Computer ◽  
Reconstruction Process ◽  
Computing Power ◽  
Order Of Magnitude

Obtaining the three-dimensional (3D) structure of negatively stained biological specimens at a resolution of, typically, 2 - 4 nm is becoming a relatively common practice in an increasing number of laboratories. A combination of new conceptual approaches, new software tools, and faster computers have made this situation possible. However, all these 3D reconstruction processes are quite computer intensive, and the middle term future is full of suggestions entailing an even greater need of computing power. Up to now all published 3D reconstructions in this field have been performed on conventional (sequential) computers, but it is a fact that new parallel computer architectures represent the potential of order-of-magnitude increases in computing power and should, therefore, be considered for their possible application in the most computing intensive tasks.We have studied both shared-memory-based computer architectures, like the BBN Butterfly, and local-memory-based architectures, mainly hypercubes implemented on transputers, where we have used the algorithmic mapping method proposed by Zapata el at. In this work we have developed the basic software tools needed to obtain a 3D reconstruction from non-crystalline specimens (“single particles”) using the so-called Random Conical Tilt Series Method. We start from a pair of images presenting the same field, first tilted (by ≃55°) and then untilted. It is then assumed that we can supply the system with the image of the particle we are looking for (ideally, a 2D average from a previous study) and with a matrix describing the geometrical relationships between the tilted and untilted fields (this step is now accomplished by interactively marking a few pairs of corresponding features in the two fields). From here on the 3D reconstruction process may be run automatically.

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